Correct option 3: South - West

South-West

Analyzing Paths and Directions

This problem involves tracing the paths of two different individuals starting from different points and meeting at a common point P. We need to determine the directional relationship between two specific points, E and Q, which are part of these paths.

Tracing the First Person's Path

Let's trace the path of the first person starting from point B. We can use a simple coordinate system to keep track of their position, assuming East is along the positive x-axis and North is along the positive y-axis.

• Starts at Point B. Let's assume B is at (0,0) for simplicity in tracking relative movement.

• Walks 4km East: Position is now (0+4, 0) = (4,0).

• Takes a right turn (South) and walks 3km to reach point C: Position is now (4, 0-3) = (4,-3). So, C is at (4,-3).

• From C, takes a left turn (East) of 4km: Position is now (4+4, -3) = (8,-3).

• Then a right turn (South) of 4km and reaches point D: Position is now (8, -3-4) = (8,-7). So, D is at (8,-7).

• From D, turns left (East) and walks for 4km to reach point E: Position is now (8+4, -7) = (12,-7). So, E is at (12,-7).

• At last, walks in the north direction and reaches Point P after walking 7km: Position is now (12, -7+7) = (12,0). So, P is at (12,0).

Based on this path, Point E is located at (12, -7) relative to our assumed starting point B(0,0), and Point P is at (12,0).

Tracing the Second Person's Path

Now let's trace the path of the second person starting from point A and ending at the same point P. We know P is at (12,0). We will use this information to find the coordinates of point Q.

• Starts at Point A. Let's assume A is at (x,y).

• Walks in the west direction from point A for 5km: Position is now (x-5, y).

• Turns right (North) and walks for 4km: Position is now (x-5, y+4).

• Turns right again (East) for 5km: Position is now (x-5+5, y+4) = (x, y+4).

• Took a left turn (North): Position is now (x, y+4).

• After walking for 3km in the same direction (North): Position is now (x, y+4+3) = (x, y+7).

• Took a left turn again (West) and walk for 5km to reach point Q: Position is now (x-5, y+7). So, Q is at (x-5, y+7).

• From Q, she walks 7km in the west direction to reach Point P: Position is now (x-5-7, y+7) = (x-12, y+7). This is Point P.

We know Point P is at (12,0). So, we can set the coordinates equal:

(x−12,y+7)=(12,0)(x−12,y+7)=(12,0)

Equating the x and y coordinates:

• x−12=12?x=12+12=24x−12=12?x=12+12=24

• y+7=0?y=0−7=−7y+7=0?y=0−7=−7

So, Point A is at (24,-7).

Now we can find the coordinates of Point Q:

Q is at (x−5,y+7)=(24−5,−7+7)=(19,0)(x−5,y+7)=(24−5,−7+7)=(19,0). So, Q is at (19,0).

Determining the Direction of Point E with respect to Point Q

We need to find the direction of Point E (12,-7) from Point Q (19,0).

To find the direction of E from Q, we look at the difference in their coordinates:

• Change in x-coordinate: xE−xQ=12−19=−7xE?−xQ?=12−19=−7. A negative change in x means moving in the West direction.

• Change in y-coordinate: yE−yQ=−7−0=−7yE?−yQ?=−7−0=−7. A negative change in y means moving in the South direction.

Since Point E is located to the West and to the South of Point Q, the direction of Point E with respect to Point Q is South-West.

Let's visualize the relative positions:

To get from Q (19,0) to E (12,-7), you move 7 units left (West) and 7 units down (South). This corresponds to the South-West direction.

Conclusion

Based on the path tracing and coordinate analysis, Point E is located South-West of Point Q.